Asset Pricing

By John H. Cochrane

Princeton University Press

Copyright © 2005 John H. Cochrane
All right reserved.

ISBN: 0691121370

Chapter One

Consumption-Based Model and Overview

An investor must decide how much to save and how much to consume, and what portfolio of assets to hold. The most basic pricing equation comes from the first-order condition for that decision. The marginal utility loss of consuming a little less today and buying a little more of the asset should equal the marginal utility gain of consuming a little more of the asset's payoff in the future. If the price and payoff do not satisfy this relation, the investor should buy more or less of the asset. It follows that the asset's price should equal the expected discounted value of the asset's payoff, using the investor's marginal utility to discount the payoff. With this simple idea, I present many classic issues in finance.

Interest rates are related to expected marginal utility growth, and hence to the expected path of consumption. In a time of high real interest rates, it makes sense to save, buy bonds, and then consume more tomorrow. Therefore, high real interest rates should be associated with an expectation of growing consumption.

Most importantly, risk corrections to asset prices should be driven by the covariance of asset payoffs with marginal utility and hence by the covariance of asset payoffs with consumption. Other things equal, an asset that does badly in states of nature like a recession, in which the investor feels poor and is consuming little, is less desirable than an asset that does badly in states of nature like a boom in which the investor feels wealthy and is consuming a great deal. The former asset will sell for a lower price; its price will reflect a discount for its "riskiness," and this riskiness depends on a co-variance, not a variance.

Marginal utility, not consumption, is the fundamental measure of how you feel. Most of the theory of asset pricing is about how to go from marginal utility to observable indicators. Consumption is low when marginal utility is high, of course, so consumption may be a useful indicator. Consumption is also low and marginal utility is high when the investor's other assets have done poorly; thus we may expect that prices are low for assets that covary positively with a large index such as the market portfolio. This is a Capital Asset Pricing Model. We will see a wide variety of additional indicators for marginal utility, things against which to compute a convariance in order to predict the risk-adjustment for prices.

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1.1 Basic Pricing Equation

An investor's first-order conditions give the basic consumption-based model,

[p.sub.t] = [E.sub.t] [[beta] u'([c.sub.t+1])/u'([c.sub.t]) [x.sub.t+1]].

Our basic objective is to figure out the value of any stream of uncertain cash flows. I start with an apparently simple case, which turns out to capture very general situations.

Let us find the value at time t of a payoff [x.sub.t+1]. If you buy a stock today, the payoff next period is the stock price plus dividend, [x.sub.t+1] = [p.sub.t+1] + [d.sub.t+1]. [x.sub.t+1] is a random variable: an investor does not know exactly how much he will get from his investment, but he can assess the probability of various possible outcomes. Do not confuse the payoff [x.sub.t+1] with the profit or return; [x.sub.t+1] is the value of the investment at time t + 1, without subtracting or dividing by the cost of the investment.

We find the value of this payoff by asking what it is worth to a typical investor. To do this, we need a convenient mathematical formalism to capture what an investor wants. We model investors by a utility function defined over current and future values of consumption,

U([c.sub.t], [c.sub.t+1]) = u([c.sub.t]) + [beta][E.sub.t][u([c.sub.t+1])],

where [c.sub.t] denotes consumption at date t. We often use a convenient power utility form,

u([c.sub.t]) = [1 / 1 - y] [[c.sup.1-y.sub.t].

The limit as y [right arrow] 1 is

u(c) = ln(c).

The utility function captures the fundamental desire for more consumption, rather than posit a desire for intermediate objectives such as mean and variance of portfolio returns. Consumption [c.sub.t+1] is also random; the investor does not know his wealth tomorrow, and hence how much he will decide to consume tomorrow. The period utility function u(7) is increasing, reflecting a desire for more consumption, and concave, reflecting the declining marginal value of additional consumption. The last bite is never as satisfying as the first.

This formalism captures investors' impatience and their aversion to risk, so we can quantitatively correct for the risk and delay of cash flows. Discounting the future by [beta] captures impatience, and [beta] is called the subjective discount factor. The curvature of the utility function generates aversion to risk and to intertemporal substitution: The investor prefers a consumption stream that is steady over time and across states of nature.

Now, assume that the investor can freely buy or sell as much of the payoff [x.sub.t+1] as he wishes, at a price [p.sub.t]. How much will he buy or sell? To find the answer, denote by e the original consumption level (if the investor bought none of the asset), and denote by [xi] the amount of the asset he chooses to buy. Then, his problem is

max u([c.sub.t]) + [E.sub.t][[beta]u([c.sub.t+1]) s.t. [xi]

[c.sub.t] + [e.sub.t] - [p.sub.t][xi],

[c.sub.t+1] = [e.sub.t+1] + [x.sub.t+1][xi].

Substituting the constraints into the objective, and setting the derivative with respect to [xi] equal to zero, we obtain the first-order condition for an optimal consumption and portfolio choice,

[p.sub.t]u'([c.sub.t]) = [E.sub.t][[beta]u'([c.sub.t+1])[x.sub.t+1]], (1.1)


[p.sub.t] = [E.sub.t][[beta] u' ([c.sub.t+1])/u'([c.sub.t]) [x.sub.t+1]]. (1.2)

The investor buys more or less of the asset until this first-order condition holds.

Equation (1.1) expresses the standard marginal condition for an optimum: [p.sub.t]u'([c.sub.t]) is the loss in utility if the investor buys another unit of the asset; [E.sub.t][[beta]u'{[c.sub.t+1])[x.sub.t+1]] is the increase in (discounted, expected) utility he obtains from the extra payoff at t+1. The investor continues to buy or sell the asset until the marginal loss equals the marginal gain.

Equation (1.2) is the central asset pricing formula. Given the payoff [x.sub.t+1] and given the investor's consumption choice [c.sub.t], [c.sub.t+1], it tells you what market price [p.sub.t] to expect. Its economic content is simply the first-order conditions for optimal consumption and portfolio formation. Most of the theory of asset pricing just consists of specializations and manipulations of this formula.

We have stopped short of a complete solution to the model, i.e., an expression with exogenous items on the right-hand side. We relate one endogenous variable, price, to two other endogenous variables, consumption and payoffs. One can continue to solve this model and derive the optimal consumption choice [c.sub.t], [c.sub.t+1] in terms of more fundamental givens of the model. In the model I have sketched so far, those givens are the income sequence [e.sub.t], [e.sub.t+1] and a specification of the full set of assets that the investor may buy and sell. We will in fact study such fuller solutions below. However, for many purposes one can stop short of specifying (possibly wrongly) all this extra structure, and obtain very useful predictions about asset prices from (1.2), even though consumption is an endogenous variable.

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1.2 Marginal Rate of Substitution/Stochastic Discount Factor

We break up the basic consumption-based pricing equation into

p = E(mx).

m = [beta] u'([c.sub.t+1]) / u'([c.sub.t]),

where [m.sub.t+1] is the stochastic discount factor.

A convenient way to break up the basic pricing equation (1.2) is to define the stochastic discount factor [m.sub.t+1]

[m.sub.t+1] = [beta] u'([c.sub.t+1]) / u'([c.sub.t]). (1.3)

Then, the basic pricing formula (1.2) can simply be expressed as

[p.sub.t] = [E.sub.t]([m.sub.t+1][x.sub.t+1]). (1.4)

When it is not necessary to be explicit about time subscripts or the difference between conditional and unconditional expectation, I will suppress the subscripts and just write p = E(mx). The price always comes at t, the payoff at t + 1, and the expectation is conditional on time-t information.

The term stochastic discount factor refers to the way m generalizes standard discount factor ideas. If there is no uncertainty, we can express prices via the standard present value formula

[p.sub.t] = 1 / [R.sup.f] [x.sub.t+1], (1.5)

where [R.sup.f] is the gross risk-free rate. 1/[R.sup.f] is the discount factor. Since gross interest rates are typically greater than one, the payoff [x.sub.t+1] sells "at a discount." Riskier assets have lower prices than equivalent risk-free assets, so they are often valued by using risk-adjusted discount factors,

[[p.sup.i.sub.t] = 1 / [R.sup.i][E.sub.t]([x.sup.i.sub.t+1]).

Here, I have added the i superscript to emphasize that each risky asset i must be discounted by an asset-specific risk-adjusted discount factor 1/[R.sup.i].

In this context, equation (1.4) is obviously a generalization, and it says something deep: one can incorporate all risk corrections by defining a single stochastic discount factor-the same one for each asset-and putting it inside the expectation. [m.sub.t+1] is stochastic or random because it is not known with certainty at time t. The correlation between the random components of the common discount factor m and the asset-specific payoff [x.sup.i] generate asset-specific risk corrections.

[m.sub.t+1] is also often called the marginal rate of substitution after (1.3). In that equation, [m.sub.t+1] is the rate at which the investor is willing to substitute consumption at time t + 1 for consumption at time t. [m.sub.t+1] is sometimes also called the pricing kernel. If you know what a kernel is and you express the expectation as an integral, you can see where the name comes from. It is sometimes called a change of measure or a state-price density.

For the moment, introducing the discount factor m and breaking the basic pricing equation (1.2) into (1.3) and (1.4) is just a notational convenience. However, it represents a much deeper and more useful separation. For example, notice that p = E(mx) would still be valid if we changed the utility function, but we would have a different function connecting m to data. All asset pricing models amount to alternative ways of connecting the stochastic discount factor to data. At the same time, we will study lots of alternative expressions of p = E(mx), and we can summarize many empirical approaches by applying them to p = E(mx). By separating our models into these two components, we do not have to redo all that elaboration for each asset pricing model.

* * *

1.3 Prices, Payoffs, and Notation

The price [p.sub.t] and payoff [x.sub.t+1] seem like a very restrictive kind of security. In fact, this notation is quite general and allows us easily to accommodate many different asset pricing questions. In particular, we can cover stocks, bonds, and options and make clear that there is one theory for all asset pricing.

For stocks, the one-period payoff is of course the next price plus dividend, [x.sub.t+1] = [p.sub.t+1] + [d.sub.t+1]. We frequently divide the payoff [x.sub.t+1] by the price [p.sub.t] to obtain a gross return

[R.sub.t+1] = [x.sub.t+1] / [p.sub.t].

We can think of a return as a payoff with price one. If you pay one dollar today, the return is how many dollars or units of consumption you get tomorrow. Thus, returns obey

1 = E(mR),

which is by far the most important special case of the basic formula p = E(mx). I use capital letters to denote gross returns R, which have a numerical value like 1.05. I use lowercase letters to denote net returns r = R - 1 or log (continuously compounded) returns r = ln(R), both of which have numerical values like 0.05. One may also quote percent returns 100 x r.

Returns are often used in empirical work because they are typically stationary over time. (Stationary in the statistical sense; they do not have trends and you can meaningfully take an average. "Stationary" does not mean constant.) However, thinking in terms of returns takes us away from the central task of finding asset prices. Dividing by dividends and creating a payoff of the form

[x.sub.t+1] = (1 + [p.sub.t+1]/[d.sub.t+1]) [d.sub.t+1]/[d.sub.t]

corresponding to a price [p.sub.t]/[d.sub.t] is a way to look at prices but still to examine stationary variables.

Not everything can be reduced to a return. If you borrow a dollar at the interest rate [R.sup.f] and invest it in an asset with return R, you pay no money out-of-pocket today, and get the payoff R - [R.sup.f]. This is a payoff with a zero price, so you obviously cannot divide payoff by price to get a return. Zero price does not imply zero payoff. It is a bet in which the value of the chance of losing exactly balances the value of the chance of winning, so that no money changes hands when the bet is made. It is common to study equity strategies in which one short-sells one stock or portfolio and invests the proceeds in another stock or portfolio, generating an excess return. I denote any such difference between returns as an excess return, [R.sup.e]. It is also called a zero-cost portfolio.

In fact, much asset pricing focuses on excess returns. Our economic understanding of interest rate variation turns out to have little to do with our understanding of risk premia, so it is convenient to separate the two phenomena by looking at interest rates and excess returns separately.

We also want to think about the managed portfolios, in which one invests more or less in an asset according to some signal. The "price" of such a strategy is the amount invested at time t, say [z.sub.t], and the payoff is [z.sub.t][R.sub.t+1]. For example, a market timing strategy might make an investment in stocks proportional to the price-dividend ratio, investing less when prices are higher. We could represent such a strategy as a payoff using [z.sub.t] = a - b([p.sub.t]/[d.sub.t]).

When we think about conditioning information below, we will think of objects like [z.sub.t] as instruments.


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